Fingerprint indexing based on novel features of minutiae triplets ...

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO. 5, MAY 2003 617TABLE 1Comparison between Germain et al.’s [4] Approach and Bhanu and Tan’s ApproachesTABLE 2NIST-4 Database Used in <strong>Fingerprint</strong> Recognition Research Publishedand identification is carried out during the online stage. During theoffline stage, fingerprints in the database are processed one-by-oneto extract minutiae [1] to construct model database. During theonline stage, the query image is processed by the same procedureto extract minutiae. Indexing components are derived from thetriplets of minutiae locations and used to map the points in thefeature space to the points in the <strong>indexing</strong> space. The potentialcorrespondences between the query image and images in thedatabase are searched in a local area in the <strong>indexing</strong> space. An<strong>indexing</strong> score is computed <strong>based</strong> on the number of trianglecorrespondences and candidate hypotheses are generated. The topN ranked hypotheses are the result of our <strong>indexing</strong> algorithm.3.1 Analysis of Angle Changes Under DistortionsWithout loss of generality, we assume that one vertex, O, of thetriangle (see Fig. 1) is ð0; 0Þ, and it does not change under distortions.Since distance is invariant under translation and rotation andrelatively invariant under scale, and angles are defined in terms ofthe ratio of distance, it can be proven that angles are invariant underthese transformations. However, because of uncertainty of minutiaelocations, the location of each vertex changes independently in asmall local area in a random manner. Suppose the locations of pointsA and B are ðx 1 ; 0Þ and ðx 2 ;y 2 Þ, x 1 > 0, y 2 > 0, and x 2 2 ð1; þ1Þ.We have tan ¼ y 2 =ðx 1 x 2 Þ. Because of the uncertainty ofminutiae locations, A and B move to A 0 ðx 1 þ x 1 ; 0Þ and B 0 ðx 2 þx 2 ;y 2 þ y 2 Þ; respectively, and changes to þ ; thentan ¼ððx 1 x 2 Þy 2 y 2 ðx 1 x 2 ÞÞ=ððx 1 x 2 Þ 2 þðx 1 x 2 Þðx 1 x 2 Þþy 2 2 þ y 2y 2 Þ:Suppose j x 1 x 2 j